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G = D20.27D6order 480 = 25·3·5

10th non-split extension by D20 of D6 acting via D6/S3=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D20.27D6, D12.14D10, C60.41C23, Dic30.15C22, Q8⋊D57S3, C15⋊D88C2, C1521(C4○D8), Q83S32D5, C157Q167C2, C52C8.19D6, (C5×Q8).37D6, (C3×Q8).5D10, Q8.10(S3×D5), D205S33C2, (C4×S3).25D10, (S3×C10).14D4, C10.151(S3×D4), D12.D57C2, C30.203(C2×D4), D6.3(C5⋊D4), C57(Q8.7D6), C34(D4.8D10), C20.41(C22×S3), (C5×Dic3).40D4, C12.41(C22×D5), (S3×C20).13C22, C153C8.15C22, (C3×D20).15C22, (C5×D12).15C22, (Q8×C15).11C22, Dic3.22(C5⋊D4), C4.41(C2×S3×D5), (S3×C52C8)⋊6C2, (C3×Q8⋊D5)⋊5C2, C2.32(S3×C5⋊D4), C6.54(C2×C5⋊D4), (C5×Q83S3)⋊2C2, (C3×C52C8).11C22, SmallGroup(480,593)

Series: Derived Chief Lower central Upper central

C1C60 — D20.27D6
C1C5C15C30C60C3×D20D205S3 — D20.27D6
C15C30C60 — D20.27D6
C1C2C4Q8

Generators and relations for D20.27D6
 G = < a,b,c,d | a20=b2=d2=1, c6=a10, bab=a-1, cac-1=dad=a11, cbc-1=dbd=a15b, dcd=c5 >

Subgroups: 604 in 124 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2 [×3], C3, C4, C4 [×3], C22 [×3], C5, S3 [×2], C6, C6, C8 [×2], C2×C4 [×3], D4 [×4], Q8, Q8, D5, C10, C10 [×2], Dic3, Dic3, C12, C12, D6, D6, C2×C6, C15, C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], Dic5, C20, C20 [×2], D10, C2×C10 [×2], C3⋊C8, C24, Dic6, C4×S3, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C3×D4, C3×Q8, C5×S3 [×2], C3×D5, C30, C4○D8, C52C8, C52C8, Dic10, C4×D5, D20, C5⋊D4, C2×C20 [×2], C5×D4 [×2], C5×Q8, S3×C8, C24⋊C2, D4⋊S3, C3⋊Q16, C3×SD16, D42S3, Q83S3, C5×Dic3, Dic15, C60, C60, C6×D5, S3×C10, S3×C10, C2×C52C8, D4⋊D5, D4.D5, Q8⋊D5, C5⋊Q16, C4○D20, C5×C4○D4, Q8.7D6, C3×C52C8, C153C8, D5×Dic3, C15⋊D4, C3×D20, S3×C20, S3×C20, C5×D12, C5×D12, Dic30, Q8×C15, D4.8D10, S3×C52C8, C15⋊D8, D12.D5, C3×Q8⋊D5, C157Q16, D205S3, C5×Q83S3, D20.27D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C22×S3, C4○D8, C5⋊D4 [×2], C22×D5, S3×D4, S3×D5, C2×C5⋊D4, Q8.7D6, C2×S3×D5, D4.8D10, S3×C5⋊D4, D20.27D6

Smallest permutation representation of D20.27D6
On 240 points
Generators in S240
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20)(21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100)(101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140)(141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160)(161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180)(181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200)(201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220)(221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240)
(1 101)(2 120)(3 119)(4 118)(5 117)(6 116)(7 115)(8 114)(9 113)(10 112)(11 111)(12 110)(13 109)(14 108)(15 107)(16 106)(17 105)(18 104)(19 103)(20 102)(21 54)(22 53)(23 52)(24 51)(25 50)(26 49)(27 48)(28 47)(29 46)(30 45)(31 44)(32 43)(33 42)(34 41)(35 60)(36 59)(37 58)(38 57)(39 56)(40 55)(61 164)(62 163)(63 162)(64 161)(65 180)(66 179)(67 178)(68 177)(69 176)(70 175)(71 174)(72 173)(73 172)(74 171)(75 170)(76 169)(77 168)(78 167)(79 166)(80 165)(81 220)(82 219)(83 218)(84 217)(85 216)(86 215)(87 214)(88 213)(89 212)(90 211)(91 210)(92 209)(93 208)(94 207)(95 206)(96 205)(97 204)(98 203)(99 202)(100 201)(121 190)(122 189)(123 188)(124 187)(125 186)(126 185)(127 184)(128 183)(129 182)(130 181)(131 200)(132 199)(133 198)(134 197)(135 196)(136 195)(137 194)(138 193)(139 192)(140 191)(141 239)(142 238)(143 237)(144 236)(145 235)(146 234)(147 233)(148 232)(149 231)(150 230)(151 229)(152 228)(153 227)(154 226)(155 225)(156 224)(157 223)(158 222)(159 221)(160 240)
(1 78 149 54 90 132 11 68 159 44 100 122)(2 69 150 45 91 123 12 79 160 55 81 133)(3 80 151 56 92 134 13 70 141 46 82 124)(4 71 152 47 93 125 14 61 142 57 83 135)(5 62 153 58 94 136 15 72 143 48 84 126)(6 73 154 49 95 127 16 63 144 59 85 137)(7 64 155 60 96 138 17 74 145 50 86 128)(8 75 156 51 97 129 18 65 146 41 87 139)(9 66 157 42 98 140 19 76 147 52 88 130)(10 77 158 53 99 131 20 67 148 43 89 121)(21 216 199 116 177 226 31 206 189 106 167 236)(22 207 200 107 178 237 32 217 190 117 168 227)(23 218 181 118 179 228 33 208 191 108 169 238)(24 209 182 109 180 239 34 219 192 119 170 229)(25 220 183 120 161 230 35 210 193 110 171 240)(26 211 184 111 162 221 36 201 194 101 172 231)(27 202 185 102 163 232 37 212 195 112 173 222)(28 213 186 113 164 223 38 203 196 103 174 233)(29 204 187 104 165 234 39 214 197 114 175 224)(30 215 188 115 166 225 40 205 198 105 176 235)
(1 137)(2 128)(3 139)(4 130)(5 121)(6 132)(7 123)(8 134)(9 125)(10 136)(11 127)(12 138)(13 129)(14 140)(15 131)(16 122)(17 133)(18 124)(19 135)(20 126)(21 231)(22 222)(23 233)(24 224)(25 235)(26 226)(27 237)(28 228)(29 239)(30 230)(31 221)(32 232)(33 223)(34 234)(35 225)(36 236)(37 227)(38 238)(39 229)(40 240)(41 151)(42 142)(43 153)(44 144)(45 155)(46 146)(47 157)(48 148)(49 159)(50 150)(51 141)(52 152)(53 143)(54 154)(55 145)(56 156)(57 147)(58 158)(59 149)(60 160)(61 88)(62 99)(63 90)(64 81)(65 92)(66 83)(67 94)(68 85)(69 96)(70 87)(71 98)(72 89)(73 100)(74 91)(75 82)(76 93)(77 84)(78 95)(79 86)(80 97)(101 199)(102 190)(103 181)(104 192)(105 183)(106 194)(107 185)(108 196)(109 187)(110 198)(111 189)(112 200)(113 191)(114 182)(115 193)(116 184)(117 195)(118 186)(119 197)(120 188)(161 205)(162 216)(163 207)(164 218)(165 209)(166 220)(167 211)(168 202)(169 213)(170 204)(171 215)(172 206)(173 217)(174 208)(175 219)(176 210)(177 201)(178 212)(179 203)(180 214)

G:=sub<Sym(240)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200)(201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220)(221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240), (1,101)(2,120)(3,119)(4,118)(5,117)(6,116)(7,115)(8,114)(9,113)(10,112)(11,111)(12,110)(13,109)(14,108)(15,107)(16,106)(17,105)(18,104)(19,103)(20,102)(21,54)(22,53)(23,52)(24,51)(25,50)(26,49)(27,48)(28,47)(29,46)(30,45)(31,44)(32,43)(33,42)(34,41)(35,60)(36,59)(37,58)(38,57)(39,56)(40,55)(61,164)(62,163)(63,162)(64,161)(65,180)(66,179)(67,178)(68,177)(69,176)(70,175)(71,174)(72,173)(73,172)(74,171)(75,170)(76,169)(77,168)(78,167)(79,166)(80,165)(81,220)(82,219)(83,218)(84,217)(85,216)(86,215)(87,214)(88,213)(89,212)(90,211)(91,210)(92,209)(93,208)(94,207)(95,206)(96,205)(97,204)(98,203)(99,202)(100,201)(121,190)(122,189)(123,188)(124,187)(125,186)(126,185)(127,184)(128,183)(129,182)(130,181)(131,200)(132,199)(133,198)(134,197)(135,196)(136,195)(137,194)(138,193)(139,192)(140,191)(141,239)(142,238)(143,237)(144,236)(145,235)(146,234)(147,233)(148,232)(149,231)(150,230)(151,229)(152,228)(153,227)(154,226)(155,225)(156,224)(157,223)(158,222)(159,221)(160,240), (1,78,149,54,90,132,11,68,159,44,100,122)(2,69,150,45,91,123,12,79,160,55,81,133)(3,80,151,56,92,134,13,70,141,46,82,124)(4,71,152,47,93,125,14,61,142,57,83,135)(5,62,153,58,94,136,15,72,143,48,84,126)(6,73,154,49,95,127,16,63,144,59,85,137)(7,64,155,60,96,138,17,74,145,50,86,128)(8,75,156,51,97,129,18,65,146,41,87,139)(9,66,157,42,98,140,19,76,147,52,88,130)(10,77,158,53,99,131,20,67,148,43,89,121)(21,216,199,116,177,226,31,206,189,106,167,236)(22,207,200,107,178,237,32,217,190,117,168,227)(23,218,181,118,179,228,33,208,191,108,169,238)(24,209,182,109,180,239,34,219,192,119,170,229)(25,220,183,120,161,230,35,210,193,110,171,240)(26,211,184,111,162,221,36,201,194,101,172,231)(27,202,185,102,163,232,37,212,195,112,173,222)(28,213,186,113,164,223,38,203,196,103,174,233)(29,204,187,104,165,234,39,214,197,114,175,224)(30,215,188,115,166,225,40,205,198,105,176,235), (1,137)(2,128)(3,139)(4,130)(5,121)(6,132)(7,123)(8,134)(9,125)(10,136)(11,127)(12,138)(13,129)(14,140)(15,131)(16,122)(17,133)(18,124)(19,135)(20,126)(21,231)(22,222)(23,233)(24,224)(25,235)(26,226)(27,237)(28,228)(29,239)(30,230)(31,221)(32,232)(33,223)(34,234)(35,225)(36,236)(37,227)(38,238)(39,229)(40,240)(41,151)(42,142)(43,153)(44,144)(45,155)(46,146)(47,157)(48,148)(49,159)(50,150)(51,141)(52,152)(53,143)(54,154)(55,145)(56,156)(57,147)(58,158)(59,149)(60,160)(61,88)(62,99)(63,90)(64,81)(65,92)(66,83)(67,94)(68,85)(69,96)(70,87)(71,98)(72,89)(73,100)(74,91)(75,82)(76,93)(77,84)(78,95)(79,86)(80,97)(101,199)(102,190)(103,181)(104,192)(105,183)(106,194)(107,185)(108,196)(109,187)(110,198)(111,189)(112,200)(113,191)(114,182)(115,193)(116,184)(117,195)(118,186)(119,197)(120,188)(161,205)(162,216)(163,207)(164,218)(165,209)(166,220)(167,211)(168,202)(169,213)(170,204)(171,215)(172,206)(173,217)(174,208)(175,219)(176,210)(177,201)(178,212)(179,203)(180,214)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200)(201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220)(221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240), (1,101)(2,120)(3,119)(4,118)(5,117)(6,116)(7,115)(8,114)(9,113)(10,112)(11,111)(12,110)(13,109)(14,108)(15,107)(16,106)(17,105)(18,104)(19,103)(20,102)(21,54)(22,53)(23,52)(24,51)(25,50)(26,49)(27,48)(28,47)(29,46)(30,45)(31,44)(32,43)(33,42)(34,41)(35,60)(36,59)(37,58)(38,57)(39,56)(40,55)(61,164)(62,163)(63,162)(64,161)(65,180)(66,179)(67,178)(68,177)(69,176)(70,175)(71,174)(72,173)(73,172)(74,171)(75,170)(76,169)(77,168)(78,167)(79,166)(80,165)(81,220)(82,219)(83,218)(84,217)(85,216)(86,215)(87,214)(88,213)(89,212)(90,211)(91,210)(92,209)(93,208)(94,207)(95,206)(96,205)(97,204)(98,203)(99,202)(100,201)(121,190)(122,189)(123,188)(124,187)(125,186)(126,185)(127,184)(128,183)(129,182)(130,181)(131,200)(132,199)(133,198)(134,197)(135,196)(136,195)(137,194)(138,193)(139,192)(140,191)(141,239)(142,238)(143,237)(144,236)(145,235)(146,234)(147,233)(148,232)(149,231)(150,230)(151,229)(152,228)(153,227)(154,226)(155,225)(156,224)(157,223)(158,222)(159,221)(160,240), (1,78,149,54,90,132,11,68,159,44,100,122)(2,69,150,45,91,123,12,79,160,55,81,133)(3,80,151,56,92,134,13,70,141,46,82,124)(4,71,152,47,93,125,14,61,142,57,83,135)(5,62,153,58,94,136,15,72,143,48,84,126)(6,73,154,49,95,127,16,63,144,59,85,137)(7,64,155,60,96,138,17,74,145,50,86,128)(8,75,156,51,97,129,18,65,146,41,87,139)(9,66,157,42,98,140,19,76,147,52,88,130)(10,77,158,53,99,131,20,67,148,43,89,121)(21,216,199,116,177,226,31,206,189,106,167,236)(22,207,200,107,178,237,32,217,190,117,168,227)(23,218,181,118,179,228,33,208,191,108,169,238)(24,209,182,109,180,239,34,219,192,119,170,229)(25,220,183,120,161,230,35,210,193,110,171,240)(26,211,184,111,162,221,36,201,194,101,172,231)(27,202,185,102,163,232,37,212,195,112,173,222)(28,213,186,113,164,223,38,203,196,103,174,233)(29,204,187,104,165,234,39,214,197,114,175,224)(30,215,188,115,166,225,40,205,198,105,176,235), (1,137)(2,128)(3,139)(4,130)(5,121)(6,132)(7,123)(8,134)(9,125)(10,136)(11,127)(12,138)(13,129)(14,140)(15,131)(16,122)(17,133)(18,124)(19,135)(20,126)(21,231)(22,222)(23,233)(24,224)(25,235)(26,226)(27,237)(28,228)(29,239)(30,230)(31,221)(32,232)(33,223)(34,234)(35,225)(36,236)(37,227)(38,238)(39,229)(40,240)(41,151)(42,142)(43,153)(44,144)(45,155)(46,146)(47,157)(48,148)(49,159)(50,150)(51,141)(52,152)(53,143)(54,154)(55,145)(56,156)(57,147)(58,158)(59,149)(60,160)(61,88)(62,99)(63,90)(64,81)(65,92)(66,83)(67,94)(68,85)(69,96)(70,87)(71,98)(72,89)(73,100)(74,91)(75,82)(76,93)(77,84)(78,95)(79,86)(80,97)(101,199)(102,190)(103,181)(104,192)(105,183)(106,194)(107,185)(108,196)(109,187)(110,198)(111,189)(112,200)(113,191)(114,182)(115,193)(116,184)(117,195)(118,186)(119,197)(120,188)(161,205)(162,216)(163,207)(164,218)(165,209)(166,220)(167,211)(168,202)(169,213)(170,204)(171,215)(172,206)(173,217)(174,208)(175,219)(176,210)(177,201)(178,212)(179,203)(180,214) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20),(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100),(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140),(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160),(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180),(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200),(201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220),(221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240)], [(1,101),(2,120),(3,119),(4,118),(5,117),(6,116),(7,115),(8,114),(9,113),(10,112),(11,111),(12,110),(13,109),(14,108),(15,107),(16,106),(17,105),(18,104),(19,103),(20,102),(21,54),(22,53),(23,52),(24,51),(25,50),(26,49),(27,48),(28,47),(29,46),(30,45),(31,44),(32,43),(33,42),(34,41),(35,60),(36,59),(37,58),(38,57),(39,56),(40,55),(61,164),(62,163),(63,162),(64,161),(65,180),(66,179),(67,178),(68,177),(69,176),(70,175),(71,174),(72,173),(73,172),(74,171),(75,170),(76,169),(77,168),(78,167),(79,166),(80,165),(81,220),(82,219),(83,218),(84,217),(85,216),(86,215),(87,214),(88,213),(89,212),(90,211),(91,210),(92,209),(93,208),(94,207),(95,206),(96,205),(97,204),(98,203),(99,202),(100,201),(121,190),(122,189),(123,188),(124,187),(125,186),(126,185),(127,184),(128,183),(129,182),(130,181),(131,200),(132,199),(133,198),(134,197),(135,196),(136,195),(137,194),(138,193),(139,192),(140,191),(141,239),(142,238),(143,237),(144,236),(145,235),(146,234),(147,233),(148,232),(149,231),(150,230),(151,229),(152,228),(153,227),(154,226),(155,225),(156,224),(157,223),(158,222),(159,221),(160,240)], [(1,78,149,54,90,132,11,68,159,44,100,122),(2,69,150,45,91,123,12,79,160,55,81,133),(3,80,151,56,92,134,13,70,141,46,82,124),(4,71,152,47,93,125,14,61,142,57,83,135),(5,62,153,58,94,136,15,72,143,48,84,126),(6,73,154,49,95,127,16,63,144,59,85,137),(7,64,155,60,96,138,17,74,145,50,86,128),(8,75,156,51,97,129,18,65,146,41,87,139),(9,66,157,42,98,140,19,76,147,52,88,130),(10,77,158,53,99,131,20,67,148,43,89,121),(21,216,199,116,177,226,31,206,189,106,167,236),(22,207,200,107,178,237,32,217,190,117,168,227),(23,218,181,118,179,228,33,208,191,108,169,238),(24,209,182,109,180,239,34,219,192,119,170,229),(25,220,183,120,161,230,35,210,193,110,171,240),(26,211,184,111,162,221,36,201,194,101,172,231),(27,202,185,102,163,232,37,212,195,112,173,222),(28,213,186,113,164,223,38,203,196,103,174,233),(29,204,187,104,165,234,39,214,197,114,175,224),(30,215,188,115,166,225,40,205,198,105,176,235)], [(1,137),(2,128),(3,139),(4,130),(5,121),(6,132),(7,123),(8,134),(9,125),(10,136),(11,127),(12,138),(13,129),(14,140),(15,131),(16,122),(17,133),(18,124),(19,135),(20,126),(21,231),(22,222),(23,233),(24,224),(25,235),(26,226),(27,237),(28,228),(29,239),(30,230),(31,221),(32,232),(33,223),(34,234),(35,225),(36,236),(37,227),(38,238),(39,229),(40,240),(41,151),(42,142),(43,153),(44,144),(45,155),(46,146),(47,157),(48,148),(49,159),(50,150),(51,141),(52,152),(53,143),(54,154),(55,145),(56,156),(57,147),(58,158),(59,149),(60,160),(61,88),(62,99),(63,90),(64,81),(65,92),(66,83),(67,94),(68,85),(69,96),(70,87),(71,98),(72,89),(73,100),(74,91),(75,82),(76,93),(77,84),(78,95),(79,86),(80,97),(101,199),(102,190),(103,181),(104,192),(105,183),(106,194),(107,185),(108,196),(109,187),(110,198),(111,189),(112,200),(113,191),(114,182),(115,193),(116,184),(117,195),(118,186),(119,197),(120,188),(161,205),(162,216),(163,207),(164,218),(165,209),(166,220),(167,211),(168,202),(169,213),(170,204),(171,215),(172,206),(173,217),(174,208),(175,219),(176,210),(177,201),(178,212),(179,203),(180,214)])

51 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D4E5A5B6A6B8A8B8C8D10A10B10C···10H12A12B15A15B20A···20F20G20H20I20J24A24B30A30B60A···60F
order1222234444455668888101010···101212151520···20202020202424303060···60
size1161220223346022240101030302212···1248444···466662020448···8

51 irreducible representations

dim1111111122222222222224444448
type+++++++++++++++++++++-
imageC1C2C2C2C2C2C2C2S3D4D4D5D6D6D6D10D10D10C4○D8C5⋊D4C5⋊D4S3×D4S3×D5Q8.7D6C2×S3×D5D4.8D10S3×C5⋊D4D20.27D6
kernelD20.27D6S3×C52C8C15⋊D8D12.D5C3×Q8⋊D5C157Q16D205S3C5×Q83S3Q8⋊D5C5×Dic3S3×C10Q83S3C52C8D20C5×Q8C4×S3D12C3×Q8C15Dic3D6C10Q8C5C4C3C2C1
# reps1111111111121112224441222442

Matrix representation of D20.27D6 in GL6(𝔽241)

17700000
177640000
000100
002405100
000010
000001
,
301810000
192110000
001087100
003613300
00002400
00000240
,
12390000
12400000
001000
000100
000001
00002401
,
641130000
641770000
001000
000100
00002401
000001

G:=sub<GL(6,GF(241))| [177,177,0,0,0,0,0,64,0,0,0,0,0,0,0,240,0,0,0,0,1,51,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[30,19,0,0,0,0,181,211,0,0,0,0,0,0,108,36,0,0,0,0,71,133,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[1,1,0,0,0,0,239,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,240,0,0,0,0,1,1],[64,64,0,0,0,0,113,177,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,1,1] >;

D20.27D6 in GAP, Magma, Sage, TeX

D_{20}._{27}D_6
% in TeX

G:=Group("D20.27D6");
// GroupNames label

G:=SmallGroup(480,593);
// by ID

G=gap.SmallGroup(480,593);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,254,100,675,185,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^20=b^2=d^2=1,c^6=a^10,b*a*b=a^-1,c*a*c^-1=d*a*d=a^11,c*b*c^-1=d*b*d=a^15*b,d*c*d=c^5>;
// generators/relations

׿
×
𝔽